Lemma 23.3.1.label Let $(X, \cm)$ be a measurable space and $f: X \to \real$, then the following are equivalent:
- (1)
$f$ is $(\cm, \cb_{\ol \real})$-measurable.
- (2)
$f$ is $(\cm, \cb_{\real})$-measurable.
Proof. (1) $\Rightarrow$ (2): $\cb_{\real}\subset \cb_{\ol \real}$.
(2) $\Rightarrow$ (1): For any $E \subset \ol{\real}$, $f^{-1}(E) = f^{-1}(E \cap \real) \in \cm$.$\square$
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